Oscillatory Motion | Causes and Effects

[Starproj]

Hi,

I have been working on the solution to a damped, sinusoidally driven system and their electric-circuit analogs. I can break the equation of motion into the homogeneous and particular portions, and understand that x(t) is the sum of the two solutions. I also understand that the homogeneous solution is the rate of decay and that the particular solution is the steady state portion. However, I can't resolve why the homogeneouos portion can be totally ignored when calculating motion or even the current. Even though it is decaying, isn't that part of the solution part of the reality of the system in question?

I am looking at Marion and Thornton "Classical Dynamics," 4th edition, sections 3.6 - 3.8.

I hope my question makes sense. I appreciate any input anyone can provide.

 

[HallsofIvy]

I don't have that book but I don't believe it says "the homogeneouos portion can be totally ignored"- that just isn't true. It is true that the "damping" part, the exponential with exponent a negative number times t, goes to 0 fairly rapidly. In looking at the "steady state" behavior, the behavior for large t, that term can be ignored.

 

[Starproj]

Hi,

You're right. "Totally ignored" are my words. It just seems that as I look through the text and the sample problems, particularly those dealing with electric circuit analogs, there is no consideration of the damping portion.

I understand what you are saying about time being large. So once an electric circuit or even a physical system like a spring in a retarding fluid, is "activated," after a large amount of time the damping portion can be ignored? Isn't the sinusoidal driving force continually competing with the damping force?

I apologize if this is obvious and I'm just not seeing it.

 

[AUMathTutor]

No, you ask a good question.

The exponential decay is known as the transient solution, because it is only important for a little while. There is a point (usually, fairly early on) where it becomes neglible. The steady-state solution, however, continues to be important forever.

The forcing does compete with the damping, but that's already accounted for in the steady-state solution. The steady-state solution *is* the forcing with the damping.